text:
\section{Introduction}

The massive and mixing neutrinos have been confirmed by the precise measurements on the atmospheric, solar, accelerator and reactor neutrino oscillations \cite{patrignani2016}. This fact implies the need for new physics beyond the $SU(3)_c^{}\times SU(2)_L^{}\times U(1)^{}_{Y}$ standard model (SM). On the other hand, the cosmological observations have indicated that the neutrino masses should be in a sub-eV range \cite{patrignani2016}. In order to understand the small neutrino masses, people have proposed various ideas, among which the tree-level seesaw \cite{minkowski1977,yanagida1979,grs1979,ms1980} mechanism is very popular \cite{minkowski1977,yanagida1979,grs1979,ms1980,mw1980,sv1980,cl1980,lsw1981,ms1981,flhj1989,ma1998}. However, the seesaw will not be easy to verify unless it is not at a naturally high scale. Alternatively, the neutrinos can acquire their tiny masses at loop order \cite{zee1980,zee1985,babu1988,bl2001,knt2003,cs2004,ma2006,lln2009,kss2011,acmn2014,amn2014,addh2015,hhop2015,ma2015,lg2015,amn2016,ammn2016,noo2016-3,no2016,no2016-2,gms2016,noo2016,ghl2016,noo2016-2,cgn2006,cgnw2006,cg2010,dabsw2012,gnr2013,cght2013,gnr2014,ght2014,gt2016}. These models for the radiative neutrino mass generation contain additionally charged scalars so that they may be tested at colliders.

In principle the neutrinos can have a Majorana nature \cite{majorana1937} since they do not carry any electric charges. One hence can expect a neutrinoless double beta decay ($0\nu\beta\beta$) \cite{furry1939} process mediated by the Majorana electron neutrinos. This $0\nu\beta\beta$ process is determined by one unknown parameter $m_{ee}^{}$, i.e. the $1\!-\!1$ element in the Majorana neutrino mass matrix, so that it can be seen in the running and planning experiments unless the $m_{ee}^{}$ parameter is big enough \cite{rodejohann2011,bgmr2013}. However, there are other possibilities for a $0\nu\beta\beta$ process \cite{ms1980,ms1981,tnnsv2010,pp2012,ddkv2012,app2013,br2013,hl2014,dgpss2015,bgm2015,glp2015,abm2015,bbgm2015}. For example, some left-right symmetric models for a linear seesaw of tree-level neutrino mass generation can of\mbox{}fer a nonconventional tree-level $0\nu\beta\beta$ process with an testable lifetime, which simply depends on the scale of the left-right symmetry breaking rather than the details of the Majorana neutrino mass matrix \cite{gu2015}. One may also consider other models which accommodate an observable $0\nu\beta\beta$ process at tree level and then give a negligible contribution to the neutrino masses at loop order \cite{sv1982}. These $0\nu\beta\beta$ processes, which are related to quite a few arbitrary parameters, thus can be free of the constraint from the neutrino mass matrix \cite{bm2002,gu2011,bhow2013}.

It should be interesting if an enhanced $0\nu\beta\beta$ process originates from a tiny $m_{ee}^{}$. Some people have realized this scenario \cite{cgn2006,cgnw2006,cg2010,dabsw2012,gnr2013,cght2013,gnr2014,ght2014,gt2016}. In a realistic model \cite{cgn2006}, after the SM Higgs doublet develops a vacuum expectation value (VEV) for spontaneously breaking the electroweak symmetry, a Higgs triplet without any Yukawa couplings can acquire an induced VEV up to a few GeV, meanwhile, its doubly charged component can mix with a doubly charged scalar singlet. Thanks to the gauge interactions, a two-loop induced Majorana neutrino mass matrix then can have a structure fully determined by the symmetric Yukawa couplings of the doubly charged scalar singlet to the right-handed leptons. As for the $0\nu\beta\beta$ process, it can appear at tree level through the same Yukawa interactions and the related gauge interactions.

In this paper we shall extend some two Higgs doublet models \cite{bflrss2012}, where the Yukawa couplings for the charged fermion mass generation only involve one Higgs scalar, to generate the required neutrino masses and the enhanced $0\nu\beta\beta$ process. Specifically we shall introduce two scalar singlets among which one carries a singly electric charge while the other one carries a doubly electric charge. The singly charged scalar without any Yukawa couplings has a cubic term with the two Higgs scalars. The doubly charged scalar has the Yukawa couplings with the right-handed leptons, besides its trilinear coupling with the singly charged scalar. After the electroweak symmetry breaking, we can obtain a dominant Majorana neutrino mass matrix at two-loop and a negligible one at three-loop level. The symmetric Yukawa couplings of the doubly charged scalar to the right-handed leptons can fully determine the structure of the neutrino mass matrix. The $0\nu\beta\beta$ processes can be induced at tree, one-loop and two-loop level. The amplitudes of these $0\nu\beta\beta$ processes are all proportional to the electron neutrino mass. The one-loop $0\nu\beta\beta$ process can arrive at an observable level even if the electron neutrino mass is extremely small. We will also study other experimental constraints and implications including some rare processes and Higgs phenomenology.

\section{The models}

In the fermion sector, the quarks and leptons are as same as the SM ones,
\begin{eqnarray}
&&\begin{array}{l}q^{}_L(3,2,+\frac{1}{6})\end{array}\!=\!\left[\begin{array}{l}u^{}_L\\
[1mm]
d^{}_L\end{array}\right],~\begin{array}{l}d^{}_R(3,1,-\frac{1}{3})\end{array},
~\begin{array}{l}u^{}_R(3,1,+\frac{2}{3})\end{array},~\begin{array}{l}l^{}_L(1,2,-\frac{1}{2})\end{array}\!=\!\left[\begin{array}{l}\nu^{}_L\\
[1mm]
e^{}_L\end{array}\right],~\begin{array}{l}e^{}_R(1,1,-1)\end{array}\,.
\end{eqnarray}
Here and thereafter the brackets following the fields describe the transformations under the $SU(2)^{}_c\times SU(2)^{}_L\times U(1)^{}_{Y}$ gauge groups. The scalar sector contains two charged singlets,
\begin{eqnarray}
\begin{array}{l}\delta^{\pm}_{}(1,1,\pm1)\end{array},
~\begin{array}{l}\xi^{\pm\pm}_{}(1,1,\pm2)\end{array}\,,
\end{eqnarray}
besides two Higgs doublets,
\begin{eqnarray}
\begin{array}{l}\phi^{}_1(1,2,+\frac{1}{2})\end{array}\!=\!\left[\begin{array}{l}\phi^{+}_{1}\\
[2mm]
\phi^{0}_{1}\end{array}\right],
~\begin{array}{l}\phi^{}_2(1,2,+\frac{1}{2})\end{array}\!=\!\left[\begin{array}{l}\phi^{+}_{2}\\
[2mm]
\phi^{0}_{2}\end{array}\right]\,.
\end{eqnarray}

We impose a softly broken discrete symmetry $\mathbb{S}$ so that the Yukawa couplings for generating the charged fermion masses will only involve one Higgs doublet. For example, we can take $\mathbb{S}=Z_2^{}$ under which one type or three types of the right-handed fermions and one Higgs scalar carry an odd parity while the other fermions and scalars carry an even parity. Alternatively, the $\mathbb{S}$ symmetry can be a global one. For instance,  we can take $\mathbb{S}=U(1)_X^{}$ under which only one Higgs scalar is non-trivial. We further assume a softly broken lepton number, under which the doubly charged scalar carries a lepton number of two units while the Higgs scalars and the singly charged scalar do not carry any lepton numbers.

We then summarize the allowed Yukawa interactions as below,
\begin{itemize}
\item \emph{Case-1}: The right-handed up-type quarks, down-type quarks and charged leptons couple to a same Higgs doublet,
\begin{eqnarray}
\mathcal{L}_Y^{} &=&-y'^{}_u\bar{q}_L^{}\tilde{\phi}^{}_1 u_R^{}-y'^{}_d\bar{q}_L^{}\phi^{}_1 d_R^{}
-y'^{}_e\bar{l}_L^{}\phi^{}_1 e_R^{}-\frac{1}{2}f\xi^{--}_{}\bar{e}_R^{}e^{c}_R+\textrm{H.c.}~~\textrm{with}~~f=f^T_{}\,;
\end{eqnarray}
\item \emph{Case-2}: The right-handed up-type quarks and  down-type quarks couple to a same Higgs doublet,
\begin{eqnarray}
\mathcal{L}_Y^{} &=&-y'^{}_u\bar{q}_L^{}\tilde{\phi}^{}_1 u_R^{}-y'^{}_d\bar{q}_L^{}\phi^{}_1 d_R^{}
-y'^{}_e\bar{l}_L^{}\phi^{}_2 e_R^{}-\frac{1}{2}f\xi^{--}_{}\bar{e}_R^{}e^{c}_R+\textrm{H.c.}~~\textrm{with}~~f=f^T_{}\,;
\end{eqnarray}
\item \emph{Case-3}: The right-handed up-type quarks and charged leptons couple to a same Higgs doublet,
\begin{eqnarray}
\mathcal{L}_Y^{} &=&-y'^{}_u\bar{q}_L^{}\tilde{\phi}^{}_1 u_R^{}-y'^{}_d\bar{q}_L^{}\phi^{}_2 d_R^{}
-y'^{}_e\bar{l}_L^{}\phi^{}_1 e_R^{}-\frac{1}{2}f\xi^{--}_{}\bar{e}_R^{}e^{c}_R+\textrm{H.c.}~~\textrm{with}~~f=f^T_{}\,;
\end{eqnarray}
\item \emph{Case-4}: The right-handed down-type quarks and charged leptons couple to a same Higgs doublet,
\begin{eqnarray}
\mathcal{L}_Y^{} &=&-y'^{}_u\bar{q}_L^{}\tilde{\phi}^{}_1 u_R^{}-y'^{}_d\bar{q}_L^{}\phi^{}_2 d_R^{}
-y'^{}_e\bar{l}_L^{}\phi^{}_2 e_R^{}-\frac{1}{2}f\xi^{--}_{}\bar{e}_R^{}e^{c}_R+\textrm{H.c.}~~\textrm{with}~~f=f^T_{}\,.
\end{eqnarray}
\end{itemize}

We also write down the full scalar potential with the following quadratic, cubic and quartic terms,
\begin{eqnarray}
\label{potential}
V&=&\mu_1^2|\phi_1^{}|^2_{}
+\mu_2^2 |\phi_2^{}|^2_{}+\mu_{12}^2(\phi^\dagger_1\phi^{}_2+\textrm{H.c.})
+\lambda_1^{}|\phi_1^{}|^4_{}+\lambda_2^{}|\phi_2^{}|^4_{}
+\lambda_{12}^{}|\phi_1^{}|^2_{}|\phi_2^{}|^2_{}
+\lambda'^{}_{12}\phi_1^\dagger\phi_2^{}\phi_2^\dagger\phi_1^{}\nonumber\\
[2mm]&&+[\lambda''^{}_{12}(\phi_1^\dagger\phi_2^{})^2_{}+\textrm{H.c.}]+(\mu_\delta^2
+\lambda_{1\delta}^{}|\phi_1^{}|^2_{}+\lambda_{2\delta}^{}|\phi_2^{}|^2_{})|\delta|^2_{}
+\lambda_\delta^{}|\delta|^4_{}+(\mu_\xi^2
+\lambda_{1\xi}^{}|\phi_1^{}|^2_{}+\lambda_{2\xi}^{}|\phi_2^{}|^2_{})|\xi|^2_{}
+\lambda_\xi^{}|\xi|^4_{}\nonumber\\
[2mm]&&+\lambda_{\delta\xi}^{}|\delta|^2_{}|\xi|^2_{}+\frac{1}{2}\omega(\xi^{++}_{}\delta^{-}_{}\delta^{-}_{}+\textrm{H.c.})+\sigma_{12}^{}(\delta^{-}_{}\phi^T_1 i\tau_2^{}\phi_2^{}+\textrm{H.c.})\,.\end{eqnarray}
Note only the $\mu_{12}^2$-term, the $\omega$-term and the $\sigma_{12}^{}$-term can softly break the additional $\mathbb{S}$ symmetry. So, if the $\mathbb{S}$ symmetry is a global one, the $\lambda''^{}_{12}$-term should be absent.

\section{Electroweak symmetry breaking and physical scalars}

The two Higgs scalars $\phi_{1,2}^{}$ can be always rotated by
\begin{eqnarray}
\chi=\left[\begin{array}{c}\chi^{+}_{}\\
[1mm]
\chi^{0}_{}\end{array}\right]=\phi_1^{}\cos\beta-\phi_2^{}\sin\beta\,,\quad
\varphi=\left[\begin{array}{c}\varphi^{+}_{}\\
[1mm]
\varphi^{0}_{}\end{array}\right]=\phi_1^{}\sin\beta+\phi_2^{}\cos\beta\,.
\end{eqnarray}
For a proper choice of the rotation angle $\beta$, only one of the newly defined Higgs scalars $\chi$ and $\varphi$ will develop a nonzero VEV to spontaneously break the electroweak symmetry. Without loss of generality, we can denote
\begin{eqnarray}
\tan\beta=\frac{\langle\phi_1^{}\rangle}{\langle\phi_2^{}\rangle}
=\frac{v_1^{}}{v_2^{}}
~~\textrm{with}~~\langle\phi_{1}^{}\rangle=\left[\begin{array}{c}0\\
[2mm]
\frac{1}{\sqrt{2}}v^{}_{1}\end{array}\right]\,,~~\langle\phi_{2}^{}\rangle=\left[\begin{array}{c}0\\
[2mm]
\frac{1}{\sqrt{2}}v^{}_{2}\end{array}\right]\,,
\end{eqnarray}
and then take
\begin{eqnarray}
\langle\chi\rangle=0\,,\quad\langle\varphi\rangle=\left[\begin{array}{c}0\\
[2mm]
\frac{1}{\sqrt{2}}v^{}_{}\end{array}\right]~~\textrm{with}~~v=\sqrt{v_1^2+v_2^2}\simeq 246\,\textrm{GeV}\,.\quad\quad
\end{eqnarray}
This means the Higgs scalar $\varphi$ will be responsible for the electroweak symmetry breaking.

In the base with $\langle\chi\rangle=0$, we can rewrite the scalar potential to be
\begin{eqnarray}
\label{potential2}
V&=&\mu_\varphi^2|\varphi|^2_{}+M_\chi^2 |\chi|^2_{}
+\kappa_1^{}|\varphi|^4_{}+\kappa_2^{}|\chi|^4_{}
+\kappa_{3}^{}|\varphi|^2_{}|\chi|^2_{}+\kappa^{}_{4}\varphi_{}^\dagger\chi\chi_{}^\dagger\varphi
+\kappa^{}_{5}[(\varphi_{}^\dagger\chi)^2_{}+\textrm{H.c.}]+(M_\delta^2
+\kappa_{\varphi\delta}^{}|\varphi|^2_{}\nonumber\\
[2mm]&&+\kappa_{\chi\delta}^{}|\chi|^2_{}+\kappa_{\varphi\chi\delta}^{}\varphi^\dagger_{}\chi+\textrm{H.c.})|\delta|^2_{}
+\kappa_\delta^{}|\delta|^4_{}+(M_\xi^2+\kappa_{\varphi\xi}^{}|\varphi|^2_{}
+\kappa_{\chi\xi}^{}|\chi|^2_{}+\kappa_{\varphi\chi\xi}^{}\varphi^\dagger_{}\chi+\textrm{H.c.})|\xi|^2_{}+\kappa_\xi^{}|\xi|^4_{}\nonumber\\
[2mm]&&
+\kappa_{\delta\xi}^{}|\delta|^2_{}|\xi|^2_{}
+\frac{1}{2}\omega(\xi^{++}_{}\delta^{-}_{}\delta^{-}_{}+\textrm{H.c.})+\sigma(\delta^{-}_{}\varphi^T_{} i\tau_2^{}\chi+\textrm{H.c.})\,.
\end{eqnarray}
Meanwhile, the Yukawa interactions can be expanded by
\begin{itemize}
\item \emph{Case-1}:
\begin{eqnarray}
\mathcal{L}_Y^{} &=&-y^{}_u\bar{q}_L^{}\tilde{\varphi} u_R^{}-y^{}_d\bar{q}_L^{}\varphi d_R^{}
-y^{}_e\bar{l}_L^{}\varphi e_R^{}- (y^{}_u\cot\beta)\bar{q}_L^{}\tilde{\chi} u_R^{}-(y^{}_d\cot\beta) \bar{q}_L^{}\chi d_R^{}-(y^{}_e\cot\beta) \bar{l}_L^{}\chi e_R^{}\nonumber\\
[2mm]&&-\frac{1}{2}f\xi^{--}_{}\bar{e}_R^{}e^{c}_R+\textrm{H.c.}\,;
\end{eqnarray}
\item \emph{Case-2}:
\begin{eqnarray}
\mathcal{L}_Y^{} &=&-y^{}_u\bar{q}_L^{}\tilde{\varphi} u_R^{}-y^{}_d\bar{q}_L^{}\varphi d_R^{}
-y^{}_e\bar{l}_L^{}\varphi e_R^{}- (y^{}_u\cot\beta)\bar{q}_L^{}\tilde{\chi} u_R^{}-(y^{}_d\cot\beta) \bar{q}_L^{}\chi d_R^{}+(y^{}_e\tan\beta) \bar{l}_L^{}\chi e_R^{}\nonumber\\
[2mm]&&-\frac{1}{2}f\xi^{--}_{}\bar{e}_R^{}e^{c}_R+\textrm{H.c.}\,;
\end{eqnarray}
\item \emph{Case-3}:
\begin{eqnarray}
\mathcal{L}_Y^{} &=&-y^{}_u\bar{q}_L^{}\tilde{\varphi} u_R^{}-y^{}_d\bar{q}_L^{}\varphi d_R^{}
-y^{}_e\bar{l}_L^{}\varphi e_R^{}- (y^{}_u\cot\beta)\bar{q}_L^{}\tilde{\chi} u_R^{}+(y^{}_d\tan\beta) \bar{q}_L^{}\chi d_R^{}
-(y^{}_e\cot\beta) \bar{l}_L^{}\chi e_R^{}\nonumber\\
[2mm]&&-\frac{1}{2}f\xi^{--}_{}\bar{e}_R^{}e^{c}_R+\textrm{H.c.}\,;
\end{eqnarray}
\item \emph{Case-4}:
\begin{eqnarray}
\mathcal{L}_Y^{} &=&-y^{}_u\bar{q}_L^{}\tilde{\varphi} u_R^{}-y^{}_d\bar{q}_L^{}\varphi d_R^{}
-y^{}_e\bar{l}_L^{}\varphi e_R^{}-(y^{}_u\cot\beta)\bar{q}_L^{}\tilde{\chi} u_R^{}+(y^{}_d\tan\beta) \bar{q}_L^{}\chi d_R^{}
+(y^{}_e\tan\beta) \bar{l}_L^{}\chi e_R^{}\nonumber\\
[2mm]&&-\frac{1}{2}f\xi^{--}_{}\bar{e}_R^{}e^{c}_R+\textrm{H.c.}\,.
\end{eqnarray}
\end{itemize}

After the Higgs scalar $\varphi$ develops its VEV for the electroweak symmetry breaking, we can take
\begin{eqnarray}
\varphi=\left[\begin{array}{c}0\\
[2mm]
\frac{1}{\sqrt{2}}(v+h)\end{array}\right],~~\chi=\left[\begin{array}{c}\chi^{+}_{}\\
[2mm]
\frac{1}{\sqrt{2}}(\chi^0_R+i\chi^0_I)\end{array}\right]\,.
\end{eqnarray}
The scalar potential (\ref{potential2}) then should give the mass terms as follows,
\begin{eqnarray}
\label{potential3}
V\supset\frac{1}{2}m_h^2 h^2_{}+\frac{1}{2}m_{\chi^{0}_R}^2(\chi^0_R)^2_{}+\frac{1}{2}m_{\chi^{0}_I}^2(\chi^0_I)^2_{}+m_{\chi^\pm_{}}^2 \chi^{+}_{}\chi^{-}_{}+m_{\delta^\pm_{}}^2 \delta^{+}_{}\delta^{-}_{}
+m_{\chi^\pm_{}\delta^\pm_{}}^2 (\chi^{+}_{}\delta^{-}_{}+\textrm{H.c.})+m_{\xi^{\pm\pm}_{}}^2 \xi^{++}_{}\xi^{--}_{}\,,
\end{eqnarray}
with
\begin{eqnarray}
m_h^2&=&2\kappa_1^{}v^2_{}=125\,\textrm{GeV}\,,\nonumber\\
[2mm]
m_{\chi^{0}_{R}}^2&=&M_\chi^2+\left(\frac{1}{2}\kappa_3^{}+\frac{1}{2}\kappa_4^{}+\kappa_5^{}\right)v^2_{}>0\,,\nonumber\\
[2mm]
m_{\chi^{0}_{I}}^2&=&M_\chi^2+\left(\frac{1}{2}\kappa_3^{}+\frac{1}{2}\kappa_4^{}-\kappa_5^{}\right)v^2_{}>0\,,\nonumber\\
[2mm]
m_{\chi^\pm_{}}^2&=&M_\chi^2+\frac{1}{2}\kappa_3^{}v^2_{}\,,\nonumber\\
[2mm]
m_{\delta^\pm_{}}^2&=&M_\delta^2+\frac{1}{2}\kappa_{\varphi\delta}^{}v^2_{}\,,\nonumber\\
[2mm]
m_{\chi^\pm_{}\delta^\pm_{}}^2&=&-\frac{1}{\sqrt{2}}v\sigma \,,\nonumber\\
[2mm]
m_{\xi^{\pm\pm}_{}}^2&=&M_\xi^2+\frac{1}{2}\kappa_{\varphi\xi}^{}v^2_{}>0\,.
\end{eqnarray}
Now the singly charged scalars $\delta^\pm_{}$ and $\chi^\pm_{}$ mix with each other. Their mass eigenstates should be
\begin{eqnarray}
\hat{\chi}^\pm_{}&=&\chi^\pm_{}\cos\theta-\delta^\pm_{}\sin\theta ~~\textrm{with}~~
m_{\hat{\chi}^\pm_{}}^2=\frac{m_{\chi^\pm_{}}^2+m_{\delta^\pm_{}}^2+\sqrt{(m_{\chi^\pm_{}}^2-m_{\delta^\pm_{}}^2)^2_{}+2v^2_{}\sigma^2_{}}}{2}>0\,,\nonumber\\
[2mm]
\hat{\delta}^\pm_{}&=&\chi^\pm_{}\sin\theta+\delta^\pm_{}\cos\theta ~~\textrm{with}~~
m_{\hat{\delta}^\pm_{}}^2=\frac{m_{\chi^\pm_{}}^2+m_{\delta^\pm_{}}^2-\sqrt{(m_{\chi^\pm_{}}^2-m_{\delta^\pm_{}}^2)^2_{}+2v^2_{}\sigma^2_{}}}{2}>0\,,
\end{eqnarray}
where the rotation angle $\theta$ is determined by
\begin{eqnarray}
\label{anglerotation}
\tan2\theta &=& \frac{\sqrt{2}v\sigma}{m_{\chi^{\pm}_{}}^2-m_{\delta^{\pm}_{}}^2}\Rightarrow
\sin^2_{} 2\theta =\frac{2v^2_{}\sigma^2_{}}{(m_{\hat{\chi}^{\pm}_{}}^2-m_{\hat{\delta}^{\pm}_{}}^2)^2_{}}
=\frac{2\left(\frac{v^2_{}}{m_{\xi^{\pm\pm}_{}}^2}\right)\left(\frac{\sigma^2_{}}{m_{\xi^{\pm\pm}_{}}^2}\right)}{\left(\frac{m_{\hat{\chi}^{\pm}_{}}^2}{m_{\xi^{\pm\pm}_{}}^2}-\frac{m_{\hat{\delta}^{\pm}_{}}^2}{m_{\xi^{\pm\pm}_{}}^2}\right)^2_{}}\leq 1~~\textrm{for}\nonumber\\
[2mm]\Delta m_{\pm}^2&\equiv& m_{\hat{\chi}^{\pm}_{}}^2-m_{\hat{\delta}^{\pm}_{}}^2 =\sqrt{(m_{\chi^\pm_{}}^2-m_{\delta^\pm_{}}^2)^2_{}+2v^2_{}\sigma^2_{}}\geq  \sqrt{2}v |\sigma|\,.
\end{eqnarray}

The charged fermions can get their masses through the Yukawa couplings involving the Higgs scalar $\varphi$, i.e.  \cite{patrignani2016}
\begin{eqnarray}
\hat{m}_u^{}&=&\frac{v}{\sqrt{2}}\hat{y}_u^{}=\textrm{diag}\{m_u^{},~m_c^{},~m_t^{}\}\simeq\textrm{diag}\{2.2\,\textrm{MeV},~1.27\,\textrm{GeV},~173\,\textrm{GeV}\}\,,\nonumber\\
[2mm]
\hat{m}_d^{}&=&\frac{v}{\sqrt{2}}\hat{y}_d^{}=\textrm{diag}\{m_d^{},~m_s^{},~m_b^{}\}\simeq\textrm{diag}\{4.7\,\textrm{MeV},~96\,\textrm{MeV}\,,~4.18\,\textrm{GeV}\}\,,\nonumber\\
[2mm]
\hat{m}_e^{}&=&\frac{v}{\sqrt{2}}\hat{y}_e^{}=\textrm{diag}\{m_e^{},~m_\mu^{},~m_\tau^{}\}
\simeq\textrm{diag}\{0.511\,\textrm{MeV},~107\,\textrm{MeV},~1.78\,\textrm{GeV}\}\,.
\end{eqnarray}
Note the perturbation requirement $|\hat{y}'^{}_{t}|<\sqrt{4\pi}$, $|\hat{y}'^{}_{b}|<\sqrt{4\pi}$ and $|\hat{y}'^{}_{\tau}|<\sqrt{4\pi}$ will constrain,
\begin{itemize}
\item \emph{Case-1}:
\begin{eqnarray}
&&\cot\beta \lesssim 3.3~~\textrm{for}~~\cot\beta<\sqrt{\frac{2\pi v^2_{}}{m_t^2}-1}\,;
\end{eqnarray}
\item \emph{Case-2}:
\begin{eqnarray}
&&0.3\lesssim\tan\beta\lesssim 346~~\textrm{for}~~\frac{1}{\sqrt{\frac{2\pi v^2_{}}{m_t^2}-1}}<\tan\beta < \sqrt{\frac{2\pi v^2_{}}{m_\tau^2}-1}\,;
\end{eqnarray}
\item \emph{Case-3}:
\begin{eqnarray}
&&6.8\times 10^{-3}_{}\lesssim\cot\beta\lesssim 3.3~~\textrm{for}~~\frac{1}{ \sqrt{\frac{2\pi v^2_{}}{m_b^2}-1}}<\cot\beta <\sqrt{\frac{2\pi v^2_{}}{m_t^2}-1}\,;
\end{eqnarray}
\item \emph{Case-4}:
\begin{eqnarray}
&&0.3\lesssim\tan\beta\lesssim147~~\textrm{for}~~\frac{1}{\sqrt{\frac{2\pi v^2_{}}{m_t^2}-1}}<\tan\beta < \sqrt{\frac{2\pi v^2_{}}{m_b^2}-1}\,.
\end{eqnarray}
\end{itemize}
Moreover, the cubic couplings $\omega$ and $\sigma$ in the potential (\ref{potential2}) should also match the perturbation requirement. Roughly speaking, we have
\begin{eqnarray}
\omega\lesssim \textrm{max}\{M_\xi^{},~M_\delta^{}\}\,,~~\sigma\lesssim \textrm{max}\{M_\delta^{},~M_\chi^{}\}\,.
\end{eqnarray}

\section{Neutrino masses}

\begin{figure*}
\vspace{6.5cm} \epsfig{file=2loop, bbllx=6cm,
bblly=6.0cm, bburx=16cm, bbury=16cm, width=8cm, height=8cm,
angle=0, clip=0} \vspace{-9.25cm} \caption{\label{2loop}
Two-loop diagram for Majorana neutrino masses. }
\end{figure*}

\begin{figure*}
\vspace{7.5cm} \epsfig{file=3loop, bbllx=6cm,
bblly=6.0cm, bburx=16cm, bbury=16cm, width=8cm, height=8cm,
angle=0, clip=0} \vspace{-7.5cm} \caption{\label{3loop}
Three-loop diagram for Majorana neutrino masses. }
\end{figure*}

As shown in Fig. \ref{2loop}, the left-handed neutrinos can obtain a Majorana mass term at two-loop level,
\begin{eqnarray}
\mathcal{L}&\supset&-\frac{1}{2}m_\nu^{}\bar{\nu}_L^{}\nu_L^c+\textrm{H.c.}\,.
\end{eqnarray}
It is easy to see that the two-loop neutrino mass matrix should have the structure as follows,
\begin{eqnarray}
\label{numass21}
m_\nu^{\textrm{2-loop}}\propto  \hat{y}_e^{}f\hat{y}_e^{}\propto \hat{m}_e^{}f\hat{m}_e^{}\,.
\end{eqnarray}
Although the three-loop diagram given in Fig. \ref{3loop} also generates the neutrino masses,
\begin{eqnarray}
\label{numass3}
m_\nu^{\textrm{3-loop}}\propto \hat{m}_e^{}f\hat{m}_e^{}\,,
\end{eqnarray}
its contribution should be much smaller than the two-loop contribution (\ref{numass21}). So, the neutrino mass matrix can be well described by
\begin{eqnarray}
\label{numass}
m_{\nu}^{}\simeq m_\nu^{\textrm{2-loop}}\gg m_\nu^{\textrm{3-loop}}\,.
\end{eqnarray}

\begin{figure*}[htp]
 \vspace{-1cm} \centering
  \includegraphics[width=15cm]{integral2.eps}
  \caption{The two-loop integral $I_2^{}(x_1^{2},x_2^{2})$ as a function of the dif\mbox{}ference $x_2^{}-x_1^{}$ for a given $x_1^{}$. The lines from top to bottom correspond to $x_1^{}=0.1$, $x_1^{}=1$, $x_1^{}=10$ and $x_1^{}=100$, respectively. }
  \label{integral2}
\end{figure*}

We calculate
\begin{eqnarray}
\label{numass23}
m_\nu^{}=\frac{c I_2^{}\sin^2_{}2\theta}{2(16\pi^2_{})^2_{}} \hat{y}_e^{}f\hat{y}_e^{}\omega =\frac{ c I_2^{}\sin^2_{}2\theta}{2^{8}_{}\pi^4_{}}\frac{ \hat{m}_e^{}f \hat{m}_e^{}}{v^2_{}} \omega
\quad \textrm{with}\quad c=\left\{\begin{array}{ll}\cot^2_{}\beta& \textrm{in}~\emph{Case-1,3}\,,\\
[2mm]
\tan^2_{}\beta& \textrm{in}~\emph{Case-2,4}\,.
\end{array}\right.
\end{eqnarray}
Here $I_2^{}$ is given by the two-loop integral,
\begin{eqnarray}
I_2^{}\left(\frac{m_{\hat{\delta}^\pm_{}}^2}{m_{\xi^{\pm\pm}_{}}^2},\frac{m_{\hat{\chi}^\pm_{}}^2}{m_{\xi^{\pm\pm}_{}}^2}\right)\!&=&\!F_2^{}
\left(\frac{m_{\hat{\delta}^\pm_{}}^2}{m_{\xi^{\pm\pm}_{}}^2},\frac{m_{\hat{\delta}^\pm_{}}^2}{m_{\xi^{\pm\pm}_{}}^2}\right)+F_2^{}
\left(\frac{m_{\hat{\chi}^\pm_{}}^2}{m_{\xi^{\pm\pm}_{}}^2},\frac{m_{\hat{\chi}^\pm_{}}^2}{m_{\xi^{\pm\pm}_{}}^2}\right)
-2F_2^{}
\left(\frac{m_{\hat{\chi}^\pm_{}}^2}{m_{\xi^{\pm\pm}_{}}^2},\frac{m_{\hat{\delta}^\pm_{}}^2}{m_{\xi^{\pm\pm}_{}}^2}\right)\,,
\end{eqnarray}
where the function $F_2^{}(x_1^{2},x_2^{2})$ is defined by
\begin{eqnarray}
F_2^{}(x_1^{2},x_2^{2})=(16\pi^2_{})^2_{}\int\frac{d^4_{}\tilde{q}_1^{}}{(2\pi)^4_{}}\frac{d^4_{}\tilde{q}_2^{}}{(2\pi)^4_{}}\frac{/\!\!\!\tilde{q}_1^{}/\!\!\!\tilde{q}_2^{}}{\tilde{q}_1^2(\tilde{q}_1^2-x_1^{2})\tilde{q}_2^2(\tilde{q}_2^2-x_2^{2})
[(\tilde{q}_1^{}+\tilde{q}_2^{})^2_{}-1]}\,,
\end{eqnarray}
with $\tilde{q}_{1,2}^{}=q_{1,2}^{}/m_{\xi^{\pm\pm}_{}}^{}$ being the reduced momentum.

In Fig. \ref{integral2}, we show the numerical results of the two-loop integral $I_2^{}(x_1^2,x_2^2)$ as a function of the dif\mbox{}ference $ x_2^{}-x_1^{}$ for a given $x_1^{}$. The lines from top to bottom correspond to $x_1^{}=0.1$, $x_1^{}=1$, $x_1^{}=10$ and $x_1^{}=100$, respectively. For a proper parameter choice, the two-loop integral $I_2^{}$ can be of the order of $\mathcal{O}(1)$. Note that $I_2^{}(x_1^2,x_2^2)\equiv 0$ for $x_1^{}=x_2^{}$. We should also keep in mind that a bigger $x_2^{}-x_1^{}$ leads to a bigger $I_2^{}$ but a smaller $\sin^2_{}2\theta$, see Eq. (\ref{anglerotation}). The product $I_2^{}\sin^2_{}2\theta$ thus cannot be very large. We hence would fail in enhancing the neutrino masses by choosing a bigger cubic coupling $\omega$, which is not allowed to be much bigger than the charged scalar masses. In other words, the charged scalars cannot be far above the electroweak scale.

Now the neutrino mass matrix has a structure fully determined by the symmetric Yukawa couplings $f_{\alpha\beta}^{}=f_{\beta\alpha}^{}$ $(\alpha,\beta = e,\mu,\tau)$, i.e.
\begin{eqnarray}
\!\!\!\!m_{\alpha\beta}^{}&\equiv& (m_\nu^{})_{\alpha\beta}^{} =\frac{ c I_2^{}\sin^2_{}2\theta}{2^{8}_{}\pi^4_{}} \frac{\omega m_\alpha^{} m_\beta^{} }{v^2_{}}f_{\alpha\beta}^{} \,.
\end{eqnarray}
Actually we read
\begin{eqnarray}
m_{ee}^{}&=&6\times 10^{-4}\,\textrm{eV}\left(\frac{f_{ee}^{}}{\sqrt{4\pi}}\right) \left(\frac{c}{1}\right)\left(\frac{I_2^{}}{1}\right)\left(\frac{\sin^2_{} 2\theta}{1}\right)\left(\frac{\omega}{1\,\textrm{TeV}}\right) \,,\nonumber\\
[2mm]
m_{\mu\mu}^{}&=&0.1\,\textrm{eV}\left(\frac{f_{\mu\mu}^{}}{0.013}\right)\left(\frac{c}{1}\right)\left(\frac{I_2^{}}{1}\right)\left(\frac{\sin^2_{} 2\theta}{1}\right)\left(\frac{\omega}{1\,\textrm{TeV}}\right) \,,\nonumber\\
[2mm]
m_{\tau\tau}^{}&=&0.1\,\textrm{eV}\left(\frac{f_{\tau\tau}^{}}{5\times 10^{-5}_{}}\right)\left(\frac{c}{1}\right)\left(\frac{I_2^{}}{1}\right)\left(\frac{\sin^2_{} 2\theta}{1}\right)\left(\frac{\omega}{1\,\textrm{TeV}}\right) \,,\nonumber\\
[2mm]
m_{e\mu}^{}&=&m_{\mu e}^{}=0.1\,\textrm{eV}\left(\frac{f_{e\mu}^{}}{2.9}\right) \left(\frac{c}{1}\right)\left(\frac{I_2^{}}{1}\right)\left(\frac{\sin^2_{} 2\theta}{1}\right)\left(\frac{\omega}{1\,\textrm{TeV}}\right) \,,\nonumber\\
[2mm]
m_{e\tau}^{}&=&m_{\tau e}^{}=0.1\,\textrm{eV}\left(\frac{f_{e\tau}^{}}{0.17}\right)\left(\frac{c}{1}\right)\left(\frac{I_2^{}}{1}\right)\left(\frac{\sin^2_{} 2\theta}{1}\right)\left(\frac{\omega}{1\,\textrm{TeV}}\right) \,,\nonumber\\
[2mm]
m_{\mu\tau}^{}&=&m_{\tau \mu}^{}=0.1\,\textrm{eV}\left(\frac{f_{\mu\tau}^{}}{8\times 10^{-4}_{}}\right)\left(\frac{c}{1}\right)\left(\frac{I_2^{}}{1}\right)\left(\frac{\sin^2_{} 2\theta}{1}\right)\left(\frac{\omega}{1\,\textrm{TeV}}\right) \,.
\end{eqnarray}

In turn, we can parametrize the Yukawa couplings $f$ by the neutrino mass matrix,
\begin{eqnarray}
f&=&\frac{2^{8}_{}\pi^4_{}v^2_{}}{ c I_2^{} \sin^2_{}2\theta}\frac{1}{\hat{m}_e^{}}m_\nu^{} \frac{1}{\hat{m}_e^{}}\frac{1}{\omega}\,.
\end{eqnarray}

\section{Neutrinoless double beta decay}

It is well known that the electron neutrino with a Majorana mass can mediate a $0\nu\beta\beta$ process.
This most popular $0\nu\beta\beta$ picture can be described by the ef\mbox{}fective operator as below,
\begin{widetext}
\begin{eqnarray}
\label{eff2loop}
\mathcal{O}^{\textrm{2-loop}}_{}&=& \frac{g^4_{}}{2m_W^4}\bar{u}_L^{}\gamma^\mu_{}d_L^{}
 \bar{u}_L^{}\gamma^\nu_{}d_L^{}\bar{e}_L^{}\gamma_\mu^{}\gamma_\nu^{} e_L^c\frac{m_{ee}^{}}{q^2_{}}
 =\frac{1}{\Lambda_2^5}\bar{u}_L^{}\gamma^\mu_{}d_L^{}
 \bar{u}_L^{}\gamma^\nu_{}d_L^{}\bar{e}_L^{}\gamma_\mu^{}\gamma_\nu^{} e_L^c~~\textrm{with}~~
 \frac{1}{\Lambda_2^5}=16 \,G_F^2 \frac{m_{ee}^{}}{q^2_{}}\,,
\end{eqnarray}
\end{widetext}
where $q=100-200\,\textrm{MeV}$ is the transfer momentum. In the present model, the above $0\nu\beta\beta$ process actually is a two-loop ef\mbox{}fect since the neutrino masses are induced at two-loop level.

\begin{figure*}
\vspace{10cm} \epsfig{file=nuless, bbllx=5cm,
bblly=6.0cm, bburx=15cm, bbury=16cm, width=8cm, height=8cm,
angle=0, clip=0} \vspace{-6cm} \caption{\label{nuless}
Tree and one-loop diagrams for neutrinoless double beta decay. }
\end{figure*}

As shown in Figs. \ref{nuless}a, our model can also generate the other $0\nu\beta\beta$ processes at tree level. The ef\mbox{}fective operators should be
\begin{widetext}
\begin{eqnarray}
\label{efftree}
\mathcal{O}^{\textrm{tree}}_{}&=&\frac{1}{\Lambda_{01}^{5}}\bar{u}_L^{}d_R^{}\bar{u}_L^{}d_R^{}\bar{e}_R^c e_R^{}+\frac{1}{\Lambda_{02}^{5}}\bar{u}_R^{}d_L^{}\bar{u}_R^{}d_L^{}\bar{e}_R^c e_R^{}+\frac{1}{\Lambda_{03}^{5}}\bar{u}_L^{}d_R^{}\bar{u}_R^{}d_L^{}\bar{e}_R^c e_R^{}~~\textrm{with}\nonumber\\
[2mm]
\frac{1}{\Lambda_{01}^{5}}&=&\frac{1}{4}\hat{y}_d^2  f_{ee}^{} c^{}_{1}\sin^2_{}2\theta\frac{\omega}{m^{2}_{\xi^{\pm\pm}_{}}}\left(\frac{1}{m_{\hat{\chi}^{\pm}_{}}^2}-\frac{1}{m_{\hat{\delta}^{\pm}_{}}^2}\right)^2_{}
=\frac{2^7_{}\pi^4_{}}{I_2^{}}\frac{c_1^{}}{c}\frac{m_d^2}{m_e^2}\frac{m_{ee}^{}\Delta m_{\pm}^2}{m^{2}_{\xi^{\pm\pm}_{}} m_{\hat{\chi}^{\pm}_{}}^2 m_{\hat{\delta}^{\pm}_{}}^2}\nonumber\\
[2mm]
&=&2\times \left(\frac{c_1^{}}{c}\right)\left(\frac{m_d^{}}{3\,\textrm{MeV}}\right)^2_{}\left(\frac{q}{100\,\textrm{MeV}}\right)^2_{}\left(\frac{\Delta m_{\pm}^2}{1\,\textrm{TeV}^2_{}}\right)\left(\frac{1\,\textrm{TeV}}{m_{\xi^{\pm\pm}_{}}^{}}\right)^2_{}
\left(\frac{1\,\textrm{TeV}}{m_{\hat{\chi}^{\pm}_{}}^{}}\right)^2_{}\left(\frac{1\,\textrm{TeV}}{m_{\hat{\delta}^{\pm}_{}}^{}}\right)^2_{}\frac{1}{\Lambda_2^5}
\,,\nonumber\\
[2mm]
\frac{1}{\Lambda_{02}^{5}}&=&\frac{1}{4}\hat{y}_u^2  f_{ee}^{} c^{}_{2}\sin^2_{}2\theta\frac{\omega}{m^{2}_{\xi^{\pm\pm}_{}}}\left(\frac{1}{m_{\hat{\chi}^{\pm}_{}}^2}-\frac{1}{m_{\hat{\delta}^{\pm}_{}}^2}\right)^2_{}
=\frac{2^7_{}\pi^4_{}}{I_2^{}}\frac{c_2^{}}{c}\frac{m_u^2}{m_e^2}\frac{m_{ee}^{}\Delta m_{\pm}^2}{m^{2}_{\xi^{\pm\pm}_{}} m_{\hat{\chi}^{\pm}_{}}^2 m_{\hat{\delta}^{\pm}_{}}^2}\nonumber\\
[2mm]
&=&0.5\times \left(\frac{c_2^{}}{c}\right)\left(\frac{m_u^{}}{1.5\,\textrm{MeV}}\right)^2_{}\left(\frac{q}{100\,\textrm{MeV}}\right)^2_{}\left(\frac{\Delta m_{\pm}^2}{1\,\textrm{TeV}^2_{}}\right)\left(\frac{1\,\textrm{TeV}}{m_{\xi^{\pm\pm}_{}}^{}}\right)^2_{}
\left(\frac{1\,\textrm{TeV}}{m_{\hat{\chi}^{\pm}_{}}^{}}\right)^2_{}\left(\frac{1\,\textrm{TeV}}{m_{\hat{\delta}^{\pm}_{}}^{}}\right)^2_{}
\frac{1}{\Lambda_2^5}\,,\nonumber\\
[2mm]
\frac{1}{\Lambda_{03}^{5}}&=&\frac{1}{4}\hat{y}_d^{}\hat{y}_u^{}  f_{ee}^{} c^{}_{3}\sin^2_{}2\theta\frac{\omega}{m^{2}_{\xi^{\pm\pm}_{}}}\left(\frac{1}{m_{\hat{\chi}^{\pm}_{}}^2}-\frac{1}{m_{\hat{\delta}^{\pm}_{}}^2}\right)^2_{}
=\frac{2^7_{}\pi^4_{}}{I_2^{}}\frac{c_3^{}}{c}\frac{m_d^{}m_u^{}}{m_e^2}\frac{m_{ee}^{}\Delta m_{\pm}^2}{m^{2}_{\xi^{\pm\pm}_{}} m_{\hat{\chi}^{\pm}_{}}^2 m_{\hat{\delta}^{\pm}_{}}^2}\nonumber\\
[2mm]
&=&1\times \left(\frac{c_3^{}}{c}\right) \left(\frac{m_d^{}}{3\,\textrm{MeV}}\right)\left(\frac{m_u^{}}{1.5\,\textrm{MeV}}\right)\left(\frac{q}{100\,\textrm{MeV}}\right)^2_{}\left(\frac{\Delta m_{\pm}^2}{1\,\textrm{TeV}^2_{}}\right)\left(\frac{1\,\textrm{TeV}}{m_{\xi^{\pm\pm}_{}}^{}}\right)^2_{}
\left(\frac{1\,\textrm{TeV}}{m_{\hat{\chi}^{\pm}_{}}^{}}\right)^2_{}\left(\frac{1\,\textrm{TeV}}{m_{\hat{\delta}^{\pm}_{}}^{}}\right)^2_{}
\frac{1}{\Lambda_2^5}\,,
\end{eqnarray}
\end{widetext}
where the coef\mbox{}ficients $c_{1,2,3}^{}$ are defined by
\begin{eqnarray}
(c_1^{},c_2^{},c_3^{})=\left\{\begin{array}{ll}(c,c,c)& \textrm{in}~\emph{Case-1}\,,\\
[2mm]
(\frac{1}{c},\frac{1}{c},\frac{1}{c})& \textrm{in}~\emph{Case-2}\,,\\[2mm]
(\frac{1}{c},c,1)& \textrm{in}~\emph{Case-3}\,,\\[2mm]
(c,\frac{1}{c},1)& \textrm{in}~\emph{Case-4}\,.\\[2mm]
\end{array}\right.
\end{eqnarray}

\begin{figure*}[htp]
 \vspace{0cm} \centering
  \includegraphics[width=15cm]{integral1.eps}
  \caption{The one-loop integral $\tilde{I}_{1}^{}(x_1^{2},x_2^{2})\equiv I_{1R}^{}(x_1^{2},x_2^{2})$ or $\tilde{I}_{1}^{}(x_1^{2},x_2^{2})\equiv I_{1I}^{}(x_1^{2},x_2^{2})$ as a function of the dif\mbox{}ference $x_2^{}-x_1^{}$ for a given $x_1^{}$. The lines from top to bottom correspond to $x_1^{}=0.1$, $x_1^{}=1$, $x_1^{}=10$ and $x_1^{}=100$, respectively. }
  \label{integral1}
\end{figure*}

Furthermore, we can realize a $0\nu\beta\beta$ process at one-loop level. The relevant diagram is shown in Fig. \ref{nuless}b. The ef\mbox{}fective operator should be
\begin{widetext}
\begin{eqnarray}
\label{eff1loop}
\mathcal{O}^{\textrm{1-loop}}_{}&=& \frac{1}{\Lambda_1^5}\bar{u}_L^{}\gamma^\mu_{}d_L^{}
 \bar{u}_L^{}\gamma_\mu^{}d_L^{}\bar{e}_R^{}e_R^c~~\textrm{with}\nonumber\\
 [2mm]
\frac{1}{\Lambda_1^5}&=&\frac{1}{2^7_{}\pi^2}\frac{g^4_{}}{m_W^4}\frac{\omega}{m^2_{\xi^{\pm\pm}_{}}}f_{ee}^{}I_{1}^{}\sin^2_{}2\theta =64\,\pi^2_{}\,G_F^2 \frac{v^2_{}}{m_{\xi^{\pm\pm}_{}}^2}\frac{m_{ee}^{}}{m_e^2}\frac{I_1^{}}{I_2^{}}\frac{1}{c}\nonumber\\
[2mm]&=&9\times 10^4_{}\times \left(\frac{q}{100\,\textrm{MeV}}\right)^2_{} \left(\frac{1\,\textrm{TeV}}{m_{\xi^{\pm\pm}_{}}^{}}\right)^2_{}\left(\frac{I_1^{}}{1}\right)\left(\frac{1}{I_2^{}}\right)\left(\frac{1}{c}\right)
\frac{1}{\Lambda_2^5}\,.
\end{eqnarray}
\end{widetext}
Here $I_{1}^{}$ is given by the one-loop integral,
\begin{widetext}
\begin{eqnarray}
I_{1}^{}&=&I_{1R}^{}\left(\frac{m_{\hat{\chi}^\pm_{}}^2}{m_{\chi^{0}_{R}}^2},\frac{m_{\hat{\delta}^\pm_{}}^2}{m_{\chi^{0}_{R}}^2}\right)
-I_{1I}^{}\left(\frac{m_{\hat{\chi}^\pm_{}}^2}{m_{\chi^{0}_{I}}^2},\frac{m_{\hat{\delta}^\pm_{}}^2}{m_{\chi^{0}_{I}}^2}\right)\,,\nonumber\\
[2mm]
I_{1R}^{}\left(\frac{m_{\hat{\chi}^\pm_{}}^2}{m_{\chi^{0}_{R}}^2},\frac{m_{\hat{\delta}^\pm_{}}^2}{m_{\chi^{0}_{R}}^2}\right)&=&F_1^{}
\left(\frac{m_{\hat{\chi}^\pm_{}}^2}{m_{\chi^{0}_{R}}^2},\frac{m_{\hat{\chi}^\pm_{}}^2}{m_{\chi^{0}_{R}}^2}\right)
+F_1^{}
\left(\frac{m_{\hat{\delta}^\pm_{}}^2}{m_{\chi^{0}_{R}}^2},\frac{m_{\hat{\delta}^\pm_{}}^2}{m_{\chi^{0}_{R}}^2}\right)-2F_1^{}
\left(\frac{m_{\hat{\chi}^\pm_{}}^2}{m_{\chi^{0}_{R}}^2},\frac{m_{\hat{\delta}^\pm_{}}^2}{m_{\chi^{0}_{R}}^2}\right)\,,\nonumber\\
[2mm]
I_{1I}^{}\left(\frac{m_{\hat{\chi}^\pm_{}}^2}{m_{\chi^{0}_{I}}^2},\frac{m_{\hat{\delta}^\pm_{}}^2}{m_{\chi^{0}_{I}}^2}\right)&=&F_1^{}
\left(\frac{m_{\hat{\chi}^\pm_{}}^2}{m_{\chi^{0}_{I}}^2},\frac{m_{\hat{\chi}^\pm_{}}^2}{m_{\chi^{0}_{I}}^2}\right)+F_1^{}
\left(\frac{m_{\hat{\delta}^\pm_{}}^2}{m_{\chi^{0}_{I}}^2},\frac{m_{\hat{\delta}^\pm_{}}^2}{m_{\chi^{0}_{I}}^2}\right)-2F_1^{}
\left(\frac{m_{\hat{\chi}^\pm_{}}^2}{m_{\chi^{0}_{I}}^2},\frac{m_{\hat{\delta}^\pm_{}}^2}{m_{\chi^{0}_{I}}^2}\right)\,,
\end{eqnarray}
\end{widetext}
with the function $F_1^{}(a,b)$ being a double integral,
\begin{eqnarray}
F_1^{}(a,b)=\int_{0}^{1}dx\int_{0}^{1-x}dy \ln(1-x-y+ax+by)\,.
\end{eqnarray}
Clearly, $I_1^{}\equiv 0$ for $m_{\chi^0_R}^{}=m_{\chi^0_I}^{}$ or $m_{\hat{\xi}^{\pm}_{}}^{}=m_{\hat{\delta}^{\pm}_{}}^{}$. In Fig. \ref{integral1}, we show the numerical results of $\tilde{I}_{1}^{}(x_1^{2},x_2^{2})\equiv I_{1R}^{}(x_1^{2},x_2^{2})$ or $\tilde{I}_{1}^{}(x_1^{2},x_2^{2})\equiv I_{1I}^{}(x_1^{2},x_2^{2})$ as a function of the dif\mbox{}ference $ x_2^{}-x_1^{}$ for a given $x_1^{}$. The lines from top to bottom correspond to $x_1^{}=0.1$, $x_1^{}=1$, $x_1^{}=10$ and $x_1^{}=100$, respectively. The one-loop integral $I_1^{}=I_{1R}^{}-I_{1I}^{}$ can be of the order of $\mathcal{O}(1)$ for a proper parameter choice.

From Eqs. (\ref{eff2loop}), (\ref{efftree}) and (\ref{eff1loop}), we can conclude
\begin{eqnarray}
\frac{1}{\Lambda_{1}^5} >> \frac{1}{\Lambda_{01}^5} \,,~\frac{1}{\Lambda_{02}^5} \,,~\frac{1}{\Lambda_{03}^5} \,,~\frac{1}{\Lambda_{2}^5}\,,
\end{eqnarray}
for a reasonable parameter choice. The $0\nu\beta\beta$ thus should be dominated by the one-loop contribution. The lifetime is determined by \cite{hkk1996}
\begin{eqnarray}
\frac{1}{T_{1/2}^{0\nu}}&=&G^{}_{0\nu}|\eta_N^{} M_N^{}|^2_{}~~\textrm{with}~~
\eta_N^{}=\frac{16\pi^2_{}}{c}\frac{I_1^{}}{I_2^{}}\frac{m_{ee}^{} m_p^{} v^2_{}}{m_{\xi^{\pm\pm}_{}}^2 m_e^2}\,,~~
M_N^{}=\frac{m_p^{}}{m_e^{}}\left[\left(\frac{f_V^{}}{f_A^{}}\right)^2_{}M_{F,N}^{}-M_{GT,N}^{}\right]\,,
\end{eqnarray}
where $G^{}_{0\nu}$ is the phase space factor, $M_{F,N}^{}$ and $M_{GT,N}^{}$ are the nuclear matrix elements, while $f_{V}^{}\approx 1$ and $f_{A}^{}\approx 1.26$ normalize the hadronic current. For the $_{}^{136}\textrm{Xe}$ isotope with $G^{}_{0\nu}=3.56\times 10^{-14}_{}\,\textrm{yr}^{-1}_{}$, $M_{GT,N}^{}=0.058$ and $M_{F,N}^{}=-0.0203$, as well as $_{}^{76}\textrm{Ge}$ isotope with $G^{}_{0\nu}=5.77\times 10^{-15}_{}\,\textrm{yr}^{-1}_{}$, $M_{GT,N}^{}=0.113$ and $M_{F,N}^{}=-0.0407$ \cite{hkk1996}, we find
\begin{eqnarray}
\label{prediction}
T_{1/2}^{0\nu}(_{}^{136}\textrm{Xe})&=&1.25\times 10^{26}_{}\,\textrm{yr}\times \left(\frac{10^{-7}_{}\,\textrm{eV}}{|m_{ee}^{}|}\right)^2_{}\left(\frac{m_{\xi^{\pm\pm}}}{1\,\textrm{TeV}}\right)^4_{}\left(\frac{c}{1}\right)^2_{}\left(\frac{I_2^{}}{1}\right)^2
\left(\frac{1}{I_1^{}}\right)^2_{}\,,\nonumber\\
[2mm]
T_{1/2}^{0\nu}(_{}^{76}\textrm{Ge})&=&2.0\times 10^{26}_{}\,\textrm{yr}\times \left(\frac{10^{-7}_{}\,\textrm{eV}}{|m_{ee}^{}|}\right)^2_{}\left(\frac{m_{\xi^{\pm\pm}}}{1\,\textrm{TeV}}\right)^4_{}\left(\frac{c}{1}\right)^2_{}\left(\frac{I_2^{}}{1}\right)^2
\left(\frac{1}{I_1^{}}\right)^2_{}\,.
\end{eqnarray}
Currently the experimental limits are $T_{1/2}^{0\nu}(_{}^{136}\textrm{Xe})>1.07\times 10^{26}_{}\,\textrm{yr}$ from the KamLAND-Zen collaboration \cite{shirai2016} and $T_{1/2}^{0\nu}(_{}^{76}\textrm{Ge})>5.2\times 10^{25}_{}\,\textrm{yr}$ from the GERDA collaboration \cite{agostini2016}. The $0\nu\beta\beta$ half-life sensitivity is expected to improve in the future, such as $T_{1/2}^{0\nu}(_{}^{136}\textrm{Xe})>8\times 10^{26}_{}\,\textrm{yr}$ \cite{auger2012} and $T_{1/2}^{0\nu}(_{}^{76}\textrm{Ge})>6\times 10^{27}_{}\,\textrm{yr}$ \cite{ackermann2013,abgrall2013}.

\section{Other constraints and implications}

The Yukawa couplings of the doubly charged scalar $\xi^{\pm\pm}_{}$ to the right-handed leptons $(e_{R}^{\pm},\,\mu_R^{\pm},\,\tau_R^{\pm})$ can result in other experimental implications \cite{nops2008,hnrs2014} such as $e^{+}_{}e^{-}_{}\rightarrow e^{+}_{}e^{-}_{}$, $e^{+}_{}e^{-}_{}\rightarrow \mu^{+}_{}\mu^{-}_{}$, $\mu^{-}_{}e^{+}_{} \rightarrow \mu^{+}_{}e^{-}_{}$, $\mu\rightarrow 3e$, $\mu\rightarrow e\gamma$, $(g-2)_{\mu}^{}$ and so on. By integrating out the doubly charged scalar and then using the Fierz transformation, we can easily give the ef\mbox{}fective Lagrangian for the electron-positron reactions $e^{+}_{}e^{-}_{}\rightarrow e^{+}_{}e^{-}_{}$ and $e^{+}_{}e^{-}_{}\rightarrow \mu^{+}_{}\mu^{-}_{}$, i.e.
\begin{eqnarray}
\mathcal{L}&\supset&\frac{|f_{ee}^{}|^2_{}}{8m_{\xi^{\pm\pm}_{}}^2}(\bar{e}_R^{}\gamma^\mu_{}e_R^{})( \bar{e}_R^{}\gamma_\mu^{} e_R^{})
+\frac{|f_{e\mu}^{}|^2_{}}{2m_{\xi^{\pm\pm}_{}}^2}(\bar{e}_R^{}\gamma^\mu_{}e_R^{})( \bar{\mu}_R^{}\gamma_\mu^{} \mu_R^{})\nonumber\\
[2mm]&=&\frac{2^{13}_{}\pi^{8}_{}}{c^2_{}I_2^2 \sin^4_{}2\theta}\frac{v^4_{}|m_{ee}^{}|^2_{}}{m_e^4 m_{\xi^{\pm\pm}_{}}^2\omega^2_{} }(\bar{e}_R^{}\gamma^\mu_{}e_R^{})( \bar{e}_R^{}\gamma_\mu^{} e_R^{})
+ \frac{2^{15}_{}\pi^{8}_{}}{c^2_{}I_2^2 \sin^4_{}2\theta}\frac{v^4_{}|m_{e\mu}^{}|^2_{}}{m_e^2 m_\mu^2 m_{\xi^{\pm\pm}_{}}^2\omega^2_{} }(\bar{e}_R^{}\gamma^\mu_{}e_R^{})( \bar{\mu}_R^{}\gamma_\mu^{} \mu_R^{})\,.
\end{eqnarray}
Similarly, we can give the ef\mbox{}fective Lagrangian for the muonium-antimuonium conversion $\mu^{-}_{}e^{+}_{} \rightarrow \mu^{+}_{}e^{-}_{}$ as below,
\begin{eqnarray}
\mathcal{L}\supset\frac{f_{ee}^{}f_{\mu\mu}^{\ast}}{8m_{\xi^{\pm\pm}_{}}^2}(\bar{e}_R^{}\gamma^\mu_{}\mu_R^{})( \bar{e}_R^{}\gamma_\mu^{} \mu_R^{})+\textrm{H.c.}=\frac{2^{13}_{}\pi^{8}_{}}{c^2_{}I_2^2 \sin^4_{}2\theta}\frac{v^4_{} m_{ee}^{} m_{\mu\mu}^\ast}{m_e^2 m_\mu^2 m_{\xi^{\pm\pm}_{}}^2\omega^2_{} }(\bar{e}_R^{}\gamma^\mu_{}\mu_R^{})( \bar{e}_R^{}\gamma_\mu^{} \mu_R^{})+\textrm{H.c.}\,.
 \end{eqnarray}
For the rare three-body decay $\mu\rightarrow 3e$, it can be described by
the ef\mbox{}fective Lagrangian,
\begin{eqnarray}
\mathcal{L}\supset\frac{f_{ee}^{}f_{e\mu}^{\ast}}{2m_{\xi^{\pm\pm}_{}}^2}(\bar{e}_R^{}\gamma^\mu_{}e_R^{})( \bar{e}_R^{}\gamma_\mu^{} \mu_R^{})+\textrm{H.c.}=\frac{2^{15}_{}\pi^{8}_{}}{c^2_{}I_2^2 \sin^4_{}2\theta}\frac{v^4_{} m_{ee}^{}m_{e\mu}^{\ast}}{m_e^3 m_\mu^{} m_{\xi^{\pm\pm}_{}}^2\omega^2_{} }(\bar{e}_R^{}\gamma^\mu_{}e_R^{})( \bar{e}_R^{}\gamma_\mu^{} \mu_R^{})+\textrm{H.c.}\,,
\end{eqnarray}
and then its decay width can be computed by
\begin{eqnarray}
\Gamma_{\mu\rightarrow 3 e}=\frac{|f_{e\mu}^{} f_{e e}^{}|^2_{}}{3\times 2^{12}_{}\pi^3_{}}\frac{m_\mu^5}{m_{\xi^{\pm\pm}_{}}^4}=\frac{2^{20}_{}\pi^{13}_{}}{3\,c^4_{}I_2^4 \sin^8_{}2\theta}\frac{v^8_{}m_\mu^3|m_{ee}^{}|^2_{}|m_{e\mu}^{}|^2_{}}{m_e^6 m_{\xi^{\pm\pm}_{}}^4  \omega^4_{}}\,.
\end{eqnarray}
By taking into account the SM result of the muon total decay width,
\begin{eqnarray}
\Gamma_{\mu}^{}\simeq \Gamma_{\mu\rightarrow e\bar{\nu}_e^{} \nu_\mu^{}}^{\textrm{SM}}=\frac{G_F^2 m_\mu^5}{192\,\pi^3_{}}\,,
\end{eqnarray}
we can read the branching ratio,
\begin{eqnarray}
\label{mu3e}
Br(\mu\rightarrow 3 e)=\frac{\Gamma_{\mu\rightarrow 3 e}}{\Gamma_{\mu}^{}}=\frac{2^{26}_{}\pi^{16}_{}}{c^4_{}I_2^4 \sin^8_{}2\theta}\frac{v^8_{}|m_{ee}^{}|^2_{}|m_{e\mu}^{}|^2_{}}{G_F^2 m_e^6 m_\mu^{2} m_{\xi^{\pm\pm}_{}}^4 \omega^4_{} }\,.
\end{eqnarray}
Clearly, we can get the similar formula for the other rare three-body decays $\tau^{-}_{}\rightarrow 3\mu^{}$, $\mu^{+}_{}\mu^{-}_{}e^{-}_{}$, $e^{+}_{}\mu^{-}_{}\mu^{-}_{}$, $\mu^{-}_{} e^{+}_{}e^{-}_{}$, $\mu^{+}_{}e^{-}_{}e^{-}_{}$, $3e$ by replacing the related parameters $m_{\alpha\beta}^{}$ in Eq. (\ref{mu3e}). We also consider the lepton flavor changing decay $\mu \rightarrow e \gamma$. The decay width should be
\begin{eqnarray}
\Gamma_{\mu\rightarrow e\gamma}&=&\frac{\alpha m_\mu^5}{9\times 2^{12}_{}\,\pi^3_{}}\frac{|f_{\mu e}^{} f_{e e}^{}|^2_{}+|f_{\mu e}^{} f_{\mu\mu}^{}|^2_{}+4|f_{\mu \tau}^{} f_{e \tau}^{}|^2_{}}{m_{\xi^{\pm\pm}_{}}^4}\nonumber\\
[2mm]&=&\frac{2^{20}_{}\pi^{13}_{}\alpha}{9\,c^4_{}I_2^4 \sin^8_{}2\theta}\frac{v^8_{}}{ m_{\xi^{\pm\pm}_{}}^4  \omega^4_{}}\left(\frac{m_\mu^3|m_{ee}^{}|^2_{}|m_{e\mu}^{}|^2_{}}{m_e^6}+
\frac{|m_{e\mu}^{}|^2_{}|m_{\mu\mu}^{}|^2_{}}{m_e^2 m_\mu^{}}+4\frac{m_\mu^3|m_{e\tau}^{}|^2_{}|m_{\mu\tau}^{}|^2_{}}{m_e^2 m_\tau^4}\right)\,,
\end{eqnarray}
and then the branching ratio,
\begin{eqnarray}
Br(\mu\rightarrow e\gamma)=\frac{\Gamma_{\mu\rightarrow e \gamma}}{\Gamma_{\mu}^{}}=\frac{2^{26}_{}\pi^{16}_{}\alpha}{3\,c^4_{}I_2^4 \sin^8_{}2\theta}\frac{v^8_{}}{G_F^2 m_{\xi^{\pm\pm}_{}}^4 \omega^4_{} }\left(\frac{|m_{ee}^{}|^2_{}|m_{e\mu}^{}|^2_{}}{m_e^6 m_\mu^2}+
\frac{|m_{e\mu}^{}|^2_{}|m_{\mu\mu}^{}|^2_{}}{m_e^2 m_\mu^{6}}+4\frac{|m_{e\tau}^{}|^2_{}|m_{\mu\tau}^{}|^2_{}}{m_e^2 m_\mu^2 m_\tau^4}\right)\,.
\end{eqnarray}
For simplicity, we do not show the similar formula for the other lepton flavor changing decays $\tau \rightarrow \mu \gamma$, $e\gamma$. We then calculate the muon anomalous magnetic momnet $(g-2)_{\mu}^{}$, i.e.
\begin{eqnarray}
\Delta a_\mu^{}=-\frac{m_\mu^2}{96\pi^2_{}}\frac{4|f_{\mu e}^{}|^2_{}+|f_{\mu\mu}^{}|^2_{}+4|f_{\mu\tau}^{}|^2_{}}{m_{\xi^{\pm\pm}_{}}^2}
=-\frac{2^{11}_{}\pi^{6}_{}}{3\,c^2_{}I_2^2 \sin^4_{}2\theta}\frac{v^4_{}}{ m_{\xi^{\pm\pm}_{}}^2 \omega^2_{}  }\left(4\frac{|m_{e\mu}^{}|^2_{}}{m_e^2}+
\frac{|m_{\mu\mu}^{}|^2_{}}{m_\mu^{2}}+4\frac{|m_{\mu\tau}^{}|^2_{}}{m_\tau^2}\right)\,.
\end{eqnarray}
We have checked that our model can escape from the above experimental constraints \cite{patrignani2016} for the parameter choice $c=\mathcal{O}(1)$, $I_2^{}=\mathcal{O}(1)$, $\sin 2 \theta\leq 1$, $m_{\xi^{\pm\pm}_{}}^{}=\mathcal{O}(\textrm{TeV})$, $\omega =\mathcal{O}( \textrm{TeV})$, $m_{e\mu,e\tau,\mu\mu,\mu\tau,\tau\tau}^{}=\mathcal{O}(0.1\,\textrm{eV})$ and $m_{ee}^{}\lesssim\mathcal{O}(10^{-4}_{}\,\textrm{eV})$.

\begin{figure*}
\vspace{8.5cm} \epsfig{file=higgs, bbllx=5cm,
bblly=6.0cm, bburx=15cm, bbury=16cm, width=7cm, height=7cm,
angle=0, clip=0} \vspace{-3cm} \caption{\label{higgs}
One-loop diagrams for giving a dimension-6 term of the SM Higgs scalar.}
\end{figure*}

We further consider the couplings of the non-SM scalars including the doublet $\chi$ as well as the singlets $\delta^{\pm}_{}$ and $\xi^{\pm\pm}_{}$ to the SM Higgs doublet $\varphi$. As shown in Fig. \ref{higgs}, these non-SM scalars can mediate some one-loop diagrams to give a dimension-6 operator \cite{zhang1993} of the SM Higgs doublet $\varphi$. The Higgs potential thus should be
\begin{eqnarray}
\label{dim6}
 V &\supset&\mu_\varphi^2\varphi^\dagger_{}\varphi+\kappa_1^{}(\varphi^\dagger_{}\varphi)^2_{}+ \frac{1}{\Lambda^2_{6}}(\varphi^\dagger_{}\varphi)^3_{}~~\textrm{with}\nonumber\\
[2mm]
\frac{1}{\Lambda^2_{6}}&=&\frac{1}{16\pi^2_{}}\left\{\frac{\kappa_3^3+(\kappa_3^{}+\kappa_4^{})^3_{}+(\kappa_3^{}+\kappa_4^{})|\kappa_5^{}|^2_{}}{M_\chi^2}+\frac{\kappa_{\varphi\delta}^{3}}{M_\delta^2}+\frac{\kappa_{\varphi\xi}^{3}}{M_\xi^2}\right.\nonumber\\
[2mm]&&\left.+
\frac{\sigma^6_{}}{(M_\chi^2-M_\delta^2)^4_{}}\left[\frac{M_\chi^4+10M_\chi^2 M_\delta^2 + M_\delta^4}{3M_\chi^2 M_\delta^2 }+2\frac{M_\chi^2+M_\delta^2}{M_\chi^2-M_\delta^2}\ln\left(\frac{M_\delta^2}{M_\chi^2}\right)\right]
\right\}\,.
\end{eqnarray}
By minimizing this potential, the quadratic and trilinear terms of the Higgs boson $h$ can be extracted,
\begin{eqnarray}
\mathcal{L}&\supset& -\frac{1}{2}m_h^2 h^2_{}-\kappa_{\textrm{ef\mbox{}f}}^{} v h^3_{}~~\textrm{with}~~m_h^2=2\kappa_1^{} v^2_{} +3 \frac{v^4_{}}{\Lambda_6^2}\,,~~\kappa_{\textrm{ef\mbox{}f}}^{}=\kappa_1^{}+\frac{5v^2_{}}{2\Lambda_6^2}
=\frac{m_h^2}{2v^2_{}}+\frac{v^2_{}}{\Lambda_6^2}\,.
\end{eqnarray}
The trilinear coupling of the Higgs boson yields a deviation from its SM value,
\begin{eqnarray}
R_{\lambda}^{}=\frac{\lambda_{\textrm{ef\mbox{}f}}^{}-\lambda_{\textrm{SM}}^{}}{\lambda_{\textrm{SM}}^{}}
=\frac{2v^4_{}}{m_h^2 \Lambda_6^2 }=0.12\left(\frac{2\,\textrm{TeV}}{\Lambda_6^{}}\right)^2_{}~~\textrm{with}~~\lambda_{\textrm{SM}}^{}=\frac{m_h^2}{2v^2_{}}\,.
\end{eqnarray}
The dimension-6 operator (\ref{dim6}) may have an interesting ef\mbox{}fect on the electroweak phase transition and hence the electroweak baryogenesis \cite{zhang1993} and may be tested at the running and future colliders\cite{mccullough2014,em2013,hgyyz2015,hjlw2016,dhot2016,zlly2016,hlw2016,klwy2016}. We now check the Higgs to diphoton decay \cite{clw2012,englert2014,cordero-cid2014,am2012,arhrib2014},
\begin{eqnarray}
R_{\gamma\gamma}^{}&\equiv& \frac{\Gamma\left(h\rightarrow \gamma\gamma\right)}{\Gamma_{\textrm{SM}}\left(h\rightarrow \gamma\gamma\right)} \nonumber\\
[2mm]
&=&\left|1+\frac{\kappa_{\hat{\chi}^{\pm}_{}}^{}v_{}^2}{2 m_{\hat{\chi}^\pm_{}}^2}\frac{A_0^{}\left(\tau_{\hat{\chi}^\pm_{}}^{}\right)}{A_1^{}\left(\tau_W^{} \right)+\frac{4}{3} A_{\frac{1}{2}}^{}\left(\tau_t^{}\right)}
+\frac{\kappa_{\hat{\delta}^{\pm}_{}}^{}v_{}^2}{2 m_{\hat{\delta}^\pm_{}}^2}\frac{A_0^{}\left(\tau_{\hat{\delta}^\pm_{}}^{}\right)}{A_1^{}\left(\tau_W^{} \right)+\frac{4}{3} A_{\frac{1}{2}}^{}\left(\tau_t^{}\right)}
+\frac{2\kappa_{\varphi\xi}^{}v_{}^2}{m_{\xi^{\pm\pm}_{}}^2}\frac{A_0^{}\left(\tau_{\xi^{\pm\pm}_{}}^{}\right)}{A_1^{}\left(\tau_W^{} \right)+\frac{4}{3} A_{\frac{1}{2}}^{}\left(\tau_t^{}\right)}
\right|^2\nonumber\\
[2mm]&&\textrm{with}~~\kappa_{\hat{\chi}^{\pm}_{}}^{}=\kappa_3^{}\cos^2_{}\theta+\kappa_{\varphi\delta}^{}\sin^2_{}\theta-\frac{\sigma\sin 2\theta }{\sqrt{2} v}\,,~~\kappa_{\hat{\delta}^{\pm}_{}}^{}=\kappa_3^{}\sin^2_{}\theta+\kappa_{\varphi\delta}^{}\cos^2_{}\theta+\frac{\sigma\sin 2\theta }{\sqrt{2} v}\,,~~\tau_X^{}=4\frac{m_X^2}{m_h^2}\,,\nonumber\\
[2mm]&&~~~~~~~\,A_{0}^{}(x)=-x^2_{}\left(\frac{1}{x}-\arcsin^2_{}\frac{1}{\sqrt{x}}\right)\,,~~A_{1}^{}(x)=-x^2_{}\left[\frac{2}{x^2_{}}+\frac{3}{x}+3\left(\frac{2}{x}-1\right)\arcsin^2_{}\frac{1}{\sqrt{x}}\right]\,,\nonumber\\
[2mm]&&~~~~~~~\,A_{\frac{1}{2}}^{}(x)=2x^2_{}\left[\frac{1}{x}+\left(\frac{1}{x}-1\right)\arcsin^2_{}\frac{1}{\sqrt{x}}\right]\,.
\end{eqnarray}
We find for $\kappa_{\hat{\chi}^{\pm}_{}}^{}=\mathcal{O}(1)$, $\kappa_{\hat{\delta}^{\pm}_{}}^{}=\mathcal{O}(1)$, $\kappa_{\varphi\xi}^{}=\mathcal{O}(1)$, $m_{\hat{\chi}^{\pm}_{}}^{}=\mathcal{O}(\textrm{TeV})$, $m_{\hat{\delta}^{\pm}_{}}^{}=\mathcal{O}(\textrm{TeV})$ and $m_{\xi^{\pm\pm}_{}}^{}=\mathcal{O}(\textrm{TeV})$, the value of $R_{\gamma\gamma}^{}$ can be much below the experimental limit \cite{patrignani2016}.

\section{Summary}

In this paper we have demonstrated the Majorana neutrino mass generation in some extended two Higgs doublet models, where the Yukawa couplings for the charged fermion mass generation only involve one Higgs scalar. In our scenario, a singly charged scalar without any Yukawa couplings has a cubic term with the two Higgs scalars. Another doubly charged scalar has the Yukawa couplings with the right-handed leptons, besides its trilinear coupling with the singly charged scalar. After the electroweak symmetry breaking, we can obtain the desired neutrino masses at two-loop level. The symmetric Yukawa couplings of the doubly charged scalar to the right-handed leptons can fully determine the structure of the neutrino mass matrix. Meanwhile, a one-loop $0\nu\beta\beta$ process can arrive at an observable level even if the electron neutrino mass is extremely tiny. We have also checked the other experimental constraints and implications including some rare processes and Higgs phenomenology.

\textbf{Acknowledgement}: The authors were supported by the Recruitment Program for Young Professionals under Grant No. 15Z127060004, the Shanghai Jiao Tong University under Grant No. WF220407201 and the Shanghai Laboratory for Particle Physics and Cosmology under Grant No. 11DZ2260700. This work was also supported by the Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education.

redpajama_set_name: RedPajamaArXiv


text:
Akrobatické lyžování na olympiádě v Calgary proběhlo jako ukázkový sport, na dalších hrách již pravidelně jako olympijský sport od roku 1992.

Program
Finále soutěží jsou vyznačeny tučně.

Pořadí zemí

Ukázkové soutěže

Muži

Ženy

Sporty na Zimních olympijských hrách 1988
1988
Akrobatické lyžování v roce 1988
Akrobatické lyžování v Kanadě

redpajama_set_name: RedPajamaWikipedia


text:
http://examiner.gmnews.com/news/2007/0607/Front_Page/004.html
Going against the status quo for cancer treatment
Healthy lifestyle changes help Millstone man win cancer battle
BY JENNIFER KOHLHEPP
MILLSTONE - When doctors couldn't agree on a method of treatment for him, Jim Matthews decided to take matters into his own hands to fight off cancer that had invaded the right half of his body.
Matthews, 46, of the Clarksburg section of Millstone, was diagnosed with cancer in 1996 at the age of 36. His daughter, Tara, was only 4 months old when doctors diagnosed him with leiomyosarcoma (or LMS), a rare type of soft-tissue sarcoma that is a neoplasm of smooth muscle.
"It's quite shocking today that one out of four people have cancer," Matthews said. "It's bound to hit close to home."
Matthews, an avid runner and a person who kept a fairly healthy lifestyle all of his life, said that six months before he was diagnosed he had pain in his right shoulder. He initially avoided seeing a doctor by icing down the pain and believing that it would eventually go away, but when the ache didn't subside, he finally caved in and made an appointment.
"I went to an orthopedic surgeon because I believed I had torn a rotator cuff," he said.
The orthopedic surgeon checked Matthews out to no avail, but Matthews soon contracted a cough on top of the shoulder pain. For the cough, he went to his family's doctor, Dr. Francis Urbanski, in the Perrineville section of Millstone.
Urbanski ran some tests and within one day the doctor told Matthews that he had discovered a large shadow on the right side of his body.
Within a week, Matthews went to the Memorial Sloan-Kettering Cancer Center in Basking Ridge, where doctors took a biopsy and later diagnosed him with malignant cancer.
Matthews said he does not know how he developed cancer or where in his body it originated.
"Using pragmatic sense, I believe it is the result of an environmental chemical, something I may have been exposed to as a kid," he said, adding that he was a golf caddy during most of his youth and spent numerous hours on greens that were sprayed with weed killers and other lawn chemicals.
"The cancer could have been in my body for 10 or 20 years, but what triggered the growth - I don't know," he said.
"Civilization has been battling cancer since the 1900s," he continued, "and some researchers believe it may be dormant in all bodies, but what spurs the growth we don't know.
"It could be diesel fumes, smoke or pesticides," he added.
Within one week of being diagnosed, Matthews said doctors performed surgery by cutting his right side open from the back and removing a grapefruit-sized tumor, his right diaphragm, at least one-third of his right lung, 3 inches of his esophagus, and - because the cancer was so close to his heart - his pericardium, which is a covering on the heart.
"The next week the doctor wanted me to go for radiation," Matthews said. "Chemotherapy was not suggested because according to statistics, radiation is the most successful at battling dormant cells."
Strong believers in health care might have felt compelled to thrust into the first medical evaluation, but Matthews said he prayed about not having to take the radiation because he had doubts.
"In the words of the Gospels and [with] all my strong Catholic upbringing, I turned my life over to Him," Matthews said.
When Matthews and his family questioned the doctor about the necessity of the radiation treatment and if it was the only option, a board of doctors at the hospital reviewed Matthews' case.
"On this board of 10 to 12 doctors, in my case when radiation was suggested, one-half said yeah and the other half said nay," Matthews said.
Due to the split vote, Matthews went out and sought various other opinions. He went to the Dana-Farber Cancer Institute in Boston, the Mayo Clinic in Rochester, N.Y., the University of Massachusetts and the University of Pennsylvania, and he got a couple of local opinions as well, he said.
"Everyone had different answers," Matthews said.
Matthews read a book by Anne E. Frahm, who had breast cancer that spread into her shoulder. Frahm, a Christian, went against what doctors prescribed and fought off cancer with nutrition. Finding the idea intriguing, Matthews discussed with his wife, Laura, the possibility of warding off cancer by juicing, taking vitamins and becoming a vegetarian. Laura ultimately convinced him to go with his heart and try that method of treatment.
"We all search out our understanding of beating cancer," he said. "Those who go traditional, as well as others that seek a road less traveled."
Matthews said that Lance Armstrong's book about his cancer experience and how radiation and chemotherapy treatments nearly killed him also played a large role in his decision to go against the status quo. He also visited with a group of cancer survivors from Orange County, N.Y., who formed Citizens Reunited to Overcome Cancer (CROC) Alumni - crocalumni.org - in 1999. Members of the nonprofit grassroots organization, whose motto is "Turning Tragedy into Triumph," led Matthews to believe that many people in fact do survive cancer while forgoing traditional treatments such as chemotherapy and radiation.
Today, Matthews is a vegetarian who regularly juice-fasts for a couple of days at a time. He only eats food grown organically, and he's a big proponent of green, leafy vegetables, Coenzyme Q10, Ester-C, Vitamin D, calcium and Vitamin A. He also advocates getting regular checkups and complete blood counts.
"As I started on my path to survival," he said, "I would guess how long it would take me to gain a certain amount of strength, and I wanted to get to five years."
He continued, "As each year went by, people would see me and would say that it didn't look like I ever had cancer."
Matthews would show those people the scars from his surgery.
As more years went by and Matthews grew even stronger during his recovery process, he decided that he wanted to reach out to other people who had been touched by cancer.
"There are groups that people can rely on for support," he said. "One of the greatest feelings of strength from the human perspective is a sense of comradeship. Trying to reach out and help others - that is what I love."
Along with finding strength in survivor organizations - and providing it as well to fellow survivors - Matthews has taken part in the local annual Relay for Life in the New Egypt section of Plumsted for the past six year. It is here where he shares in the emotions of those touched by cancer "who seek peace and solace to live each day to the best of their ability."
"When people relate to each other about survival, a very powerful feeling comes over you that carries you into the next years," he said.
Matthews, who will have been in remission for 11 years come August, said his condition is truly a mystery since he believes seeking good health played only a small part.
"So many friends I have seen leave this earth since my years in support of those with my specific cancer," he said. "The real factor I have gained is growing in my faith stronger."
Through his experience with cancer, Matthews said he has learned that the real fortune of life is family and peace in survival.
His daughter is now 11 years old, and he and his wife have two other children, Derek, 9, and Sawyer Thomas, 7.
"Having a second chance," he said. "What a blessing."

redpajama_set_name: RedPajamaCommonCrawl


text:
By Charles Derbyshire
El Filibusterismo, the second of Jose Rizal's novels of Philippinelife, is a story of the last days of the Spanish regime in the Philippines. Under the name of The Reign of Greed it is for the first time translated into English. Written some four or five years after Noli Me Tangere, the book represents Rizal's more mature judgment on political and social conditions in the islands, and in the islands, and in its graver and less hopeful tone reflects the disappointments and discouragements which he had encountered in his efforts to lead the way to reform. Rizal's dedication to the first edition is of special interest, as the writing of it was one of the grounds of accusation against him when he was condemned to death in 1896. It reads:
"To the memory of the priests, Don Mariano Gomez (85 years old), Don Jose Burgos (30 years old), and Don Jacinto Zamora (35 years old). Executed in Bagumbayan Field on the 28th of February, 1872.
"The Church, by refusing to degrade you, has placed in doubt the crime that has been imputed to you; the Government, by surrounding your trials with mystery and shadows, cause the belief that there was some error committed in fatal moments, and all the Philippines, by worshiping your memory and calling you martyrs, in no sense recognizes your culpability. In so far, therefore, as your complicity in the Cavite mutiny is not clearly proved, as you may or you may not have been patriots and as you may and may not have cherished sentiments for justice and for liberty. I have the right to dedicate my work to you as victims of the evil which I undertake to combat. And while we await expectantly upon Spain someday to restore your good name and cease to be answerable for your death, let these pages serve as a tardy wreath of dried leaves over your unknown tombs, and let it be understood that every one who without clear proofs attacks your memory stains his hands in your blood. J. Rizal."


text:
Friday Wrap-Up
The dead of Summer has not hit South Florida. After over two years in business, we could not be busier at Heitner Legal. Quick shout out to Inside The League, a publication I love, for getting on board as a partner of Sports Agent Blog. I thought it was funny that the Daily Mail decided to cite my tweet calling out Miko Grimes (NFL defensive back Brent Grimes' wife) for being disgusting after her commentary containing a reference to Jews. Off to Orlando tomorrow morning for a high school buddy's wedding. Then back to the grind in (sometimes) sunny South Florida.
This week on Forbes:
(1) This Week In Sports Law: A Messi Situation, Pete Rose Rips Accuser, Donald Sterling Stiffs Lawyers;
(2) Why WME-IMG Led $4 Billion Bid For UFC;
(3) Adidas Springs A Lawsuit On Skechers For Patent Infringement; and
(4) Tom Brady Stuck With 4-Game Suspension After Federal Court Declines To Hear Case.
This week on Inc:
(1) How a Chance Meeting With Flo Rida Launched This Licensing Agency;
(2) How This Company Mastered Productivity With 2-Hour Breaks, $1 Lunches, and Midday Workouts; and
(3) How Amanda Lee Earns $5,000 Per Instagram Post.
And as always, the weekly wrap-up:
Gary Sheffield says it wasn't the money that drew him to becoming an agent . . . he gets satisfaction out of making sure good players get what they deserve [Where I'm Coming From].
Paul Kinzer says Encarnacion thought mid-season negotiations were a distraction last time and that he's not content and happy [Encarnacion's agent explains why extension talks with the Blue Jays are on hold].
Scott Boras was vocal in explaining the decision [Matt Harvey to have surgery to address thoracic outlet syndrome].
Rob Roche said Tucker is "disillusioned" with the process right now [Agent: Justin Tucker won't re-sign if he plays on tag].
Alexei Dementiev claims "all the answers are in the Bruins office" [KHOKHLACHEV'S AGENT: ASK BRUINS WHY HE NEVER GOT A SHOT AT AN NHL JOB].
Above The Law interviews ESPN's and MMQB's Andrew Brandt [The View From Up North: Meet Andrew Brandt — NFL Exec, Player Agent, Sports Business Analyst].
Tags Alexei Dementiev, andrew brandt, gary sheffield, Paul Kinzer, Rob Roche, scott boras
By Darren Heitner
Darren Heitner created Sports Agent Blog as a New Year's Resolution on December 31, 2005. Originally titled, "I Want To Be A Sports Agent," the website was founded with the intention of causing Heitner to learn more about the profession that he wanted to join, meet reputable individuals in the space and force himself to stay on top of the latest news and trends.
Heitner now runs Heitner Legal, P.L.L.C., which is a law firm with many practice areas, including sports law and contract law. Heitner has represented numerous athletes and sports agents as legal counsel. He has also served as an Adjunct Professor at Indiana University Bloomington from 2011-2014, where he created and taught a course titled, Sport Agency Management, which included subjects ranging from NCAA regulations to athlete agent certification and the rules governing the profession. Heitner serves as an Adjunct Professor at the University of Florida Levin College of Law, where he teaches a Sports Law class that includes case law surrounding athlete agents and the NCAA rules.
← On To The Next One: Evan Turner → It's All In The Odds Who Are The Favourites For The EPL Title


text:
Q: How can I support game center for iOS 4 but still run on 3 How can I support game center for iOS 4 but still run app on iOS 3. So game center will available only on devices with iOS 4.

A: You should make a weak link between gamekit framework and your app. You can make weak link with any framework in your project's settings. Also check apple game kit programming guide for checking if game center is available. Basically it boils down to single function
BOOL isGameCenterAvailable() {
    // Check for presence of GKLocalPlayer API.
    Class gcClass = (NSClassFromString(@"GKLocalPlayer"));

    // The device must be running running iOS 4.1 or later.
    NSString *reqSysVer = @"4.1";
    NSString *currSysVer = [[UIDevice currentDevice] systemVersion];
    BOOL osVersionSupported = ([currSysVer compare:reqSysVer options:NSNumericSearch] != NSOrderedAscending);

    return (gcClass && osVersionSupported);
}

as well as including GameKit/GameKit.h header file in your prefix.pch file.

redpajama_set_name: RedPajamaStackExchange


text:
Q: How to upload Mp3 files on Qt (QMediaPlayer)? I have to create a Qt application that executes mp3 files. I use the library QMediaPlayer, how can I let the user choose and upload the mp3 file?

A: QMediaPlayer has a setMedia method which can accept a path to you local .mp3 file. It can be used like this.
QMediaPlayer *player = new QMediaPlayer;
player->setMedia(QUrl::fromLocalFile("C:\mycoolsong.mp3"));

I guess you want to build something like a file explorer from where the user can select from multiple songs. There is a great tutorial for this here: C++ Qt 49 - QFileSystemModel a simple file explorer example


Congrats to our CPP vendors that won 2016 Modern Library Awards!
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This 30-minute presentation will explain some important trends in mobile technology and clearly define the ways your library can meet those mobile demands with a library-branded app. We will give you a demonstration of how a Boopsie mobile app works and answer any questions you may have as time permits.
Boopsie, a mobile platform-as-a-service provider, builds library-branded apps that conveniently connect your community to the resources and services your library offers. Boopsie apps deliver an easy user experience and work with any ILS provider as well as all major platforms including Android, Apple, Kindle Fire and Blackberry.
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Sanlitun (三里屯 in Chinese) is located in Chaoyang, a district west of the Forbidden City not far from the Guomao area. Sanlitun includes the Taikoo Li shopping area in the north, which is full of modern and sophisticated buildings and houses many stores and shops. South of the Gongren Tiyuchang road lies the SOHO compound, with several sophisticated office, commercial and residential towers. Here there are many cafes and restaurants. Skateboarders meet in the central round square in the SOHO complex. Sanlitun attracts a young and hip crowd and many foreign tourists, due to its shopping opportunities and nightlife.


Recently, Cherry Hill — and much of the United States' eastern seaboard — was hit by Mother Nature in a way we're not entirely accustomed to seeing.
While we escaped with far less damage and destruction than many forecasters had originally predicted, Hurricane Irene tore through the region with significant force, leaving behind flooded basements and dozens of downed trees and power lines that left a large percentage of our residents without electricity for days.
This was quite a test for the township's Office of Emergency Management (OEM), which was staffed around the clock during the storm by personnel from Cherry Hill's Police and Fire departments, as well as staff from Public Works, Code Enforcement, and the mayor's office.
After monitoring the storm's movement through the course of the preceding week, OEM personnel, including my staff, began outlining Cherry Hill's emergency plans on Thursday before the storm hit. For days, our professionals worked tirelessly to monitor the ever-evolving forecast and changing conditions, and adapt our plan to respond accordingly.
Until the storm actually hit our area, it was impossible to say with total certainty what lay in store for Cherry Hill — and any level of underestimation could have proven disastrous to both private property and public safety.
From early Saturday, our operation was moving at full capacity.
DPW handed out nearly 2,000 sandbags on Saturday afternoon. Our code official spent all night inspecting flooded homes to ensure their structural integrity. Police staffed not only the OEM, but an emergency shelter at Cherry Hill East that, thankfully, only a handful of residents needed to use.
And our communications team staffed phones through the night while using Facebook and Twitter to keep residents up to speed on what was happening outside in real time.
Our efforts carried into the beginning of the week, as DPW worked to clear trees that had fallen, blocking roads and pulling down electrical lines in the process, and the Fire Department began pumping water out of basements still saddled with significant flooding.
And the cleanup continues. To help residents dig out from the piles of tree limbs and other natural debris left scattered around their properties, I have ordered special curbside collections of yard waste by our Department of Public Works.
Typically, municipal ordinance requires that yard waste — particularly branches, shrubs and small trees — be cut down into four-foot-long pieces and wrapped in bundles of no more than 50 lbs. Those regulations are being relaxed under this program, with the hope of easing the burden on homeowners.
To schedule a collection, residents need only call DPW at 424–4422. Then, place any debris at the curb line, and provided each piece is of a reasonable weight — that is, as long as it's capable of being lifted and moved — DPW will chip up the pieces on-site and haul them away. Collections will be completed as staff and equipment permit, but all requests will be filled within a few days of your phone call — we only ask that residents register by Wednesday, Sept. 14.
I have also asked DPW to be flexible with regard to the placing of water-damaged items curbside for weekly trash collection. Ordinance requires that items be put at the curb no earlier than 5 p.m. the night before collection. Through Sept. 16, those rules will be relaxed to allow homeowners to put damaged furniture and other items curbside as they're removed from the home.
Hurricane Irene has caused enough of a headache for homeowners in our area. Over the next few weeks, we will dry out, clean up, and move back toward life as usual.
In the meantime, please let my office know if there's anything we can do to make this process easier by calling us at 488–7878, or e-mailing me directly at MayorPlatt@chtownship.com.


Finnish men's goalball coach Jarkko Pokkinen and his team may have been out of the limelight since 2017 but this year they are firmly focused on making it their most successful yet.
Pokkinen's side managed to stay in the A pool when they hosted the International Blind Sports Federation (IBSA) Goalball European Championships A in Pajulahti two years ago with a seventh place finish.
This year they will compete again at the Euros in Rostock, Germany, from 5-14 October aiming for a better result which will hopefully lead to biggest stage of them all.
"After the 2017 championships, we've only played in one single tournament and had almost two years to improve our players, our skills and quality of our game. Now, we're ready to see the results of that work", Pokkinen explained. "Our main goal for this year is to win the European Championships and ultimately qualify for the Tokyo 2020 Paralympics".
Competition is strong with each of the European teams looking to secure that all important place at the Paralympic Games. Defending champions and Paralympic title holders Lithuania will compete as well as Belgium and Germany, medallists from the 2018 IBSA Goalball World Championships.
But Finland will also get another chance to qualify for Tokyo 2020, which may make their European campaign a bit more relaxed.
From 2-9 July, Fort Wayne, Indiana, USA, will host the 2019 IBSA Goalball and Judo International Qualifier, with the top two teams advancing to Tokyo 2020.
"USA are the home team and know the conditions really well, so they are probably our biggest competitor" Pokkinen said.
"Goalball is a lifestyle to me. I get inspired from loving this great sport and it's an honour to be part of the team. As well as that, the team get inspiration from our fans in Finland as in our eyes, they act as the fourth player on the court. And we're inspired by ourselves and the great team we have".
Both the Finland men and women's goalball teams have had much success in the sport, which is played all around the world. Despite this though, it can be challenging for those interested to find a way to break into the sport.
Pokkinen's players have found their way to the team through different routes and encourage anyone interested to find out if there is any club close to where they live that they can get involved in, as well as contacting support organisations for people with visual impairments. Social media also plays its part with more information and games available to view.
As well as finding a local club and support, personal dedication and hard work are another key part of the team's success. As the Pokkinen explains, "One of the barriers is personal resources, because players have to work and study to earn money to practice goalball. Our national team has good resources for the practices and competitions, but individuals have to make their living and resources in their own personal life".
That aside, and perhaps making the dedication and determination even stronger this year, the team are enjoying their journey so far and growing as a team.
The 2019 IBSA Goalball and Judo International Qualifier will see the top two men's and women's teams advance to Tokyo 2020.
The gold medallist from the 2019 IBSA Goalball European Championships will also qualify.


text:
Comparisons of IASI-A and AATSR measurements of top-of-atmosphere radiance over an extended period
Bali, M., Mittaz, J. P., Maturi, E. and Goldberg, M. D. (2016) Comparisons of IASI-A and AATSR measurements of top-of-atmosphere radiance over an extended period. Atmospheric Measurement Techniques, 9 (7). pp. 3325-3336. ISSN 1867-8548
Text (Open access) - Published Version
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To link to this item DOI: 10.5194/amt-9-3325-2016
This study examines the trustworthiness of the Advanced Along-Track Scanning Radiometer (AATSR) and the Infrared Atmospheric Sounding Interferometer (IASI-A), as on-orbit reference instruments that are useful in re-calibrating the Advanced Very High Resolution Radiometer (AVHRR) series (Mittaz and Harris, 2011). To do this, a 39-month period (1 January 2008 to 31 March 2011) of AATSR and IASI-A inter-comparisons of top-of-atmosphere (TOA) radiance measurements is examined. Our inter-comparison reveals features of the AATSR and IASI-A bias with respect to scan angle, scene temperature, time and orbital maneuvers, and gives insight into their trustworthiness as an in-orbit reference instruments. The first feature that our study reveals is that the AATSR (nadir view) and IASI-A are both stable (have no perceptible trends in the period of study). The second feature is that IASI-A is perhaps more accurate ( ∼  0.05 K) than its stated accuracy (0.5 K). In fact the AATSR and IASI-A bias is close to the AATSR pre-launch bias (plus a small offset of +0.07 K) implying that IASI-A can get close to pre-launch levels of accuracy. Third, a very small scan angular dependence of AATSR and IASI-A bias indicates that the IASI-A response vs. scan angle algorithm is robust, while the instrument is in orbit. Inter-comparisons of AATSR with IASI-A further reveal the impact of orbital maneuvers of the ENVISAT, the platform carrying AATSR, done in October 2011 and not anticipated previously. Our study reveals that this maneuver introduced a temperature-dependent bias in the AATSR measurements for low temperatures (< 240 K) in the period following this maneuver (Cocevar et al., 2011). Our study also shows that the known AATSR 12 µm channel offset is in fact temperature dependent, grows up to 0.4 K, varies seasonally and is correlated with instrument temperature and cannot be corrected by shifting the spectral response function (SRF) of AATSR. We also present a set of recommendations to help identify the parameters under which these instruments can provide the most trustworthy observations for the AVHRR re-calibration.
26 Sep 2016 09:46 Date item deposited into CentAUR


Toyota has invented this brilliant hatchback car which is known as Etios Liva it comes in both diesel and petrol versions. Toyota Etios Liva is the known to be best in class with best in class Safety, Quality, Comfort, and Reliability. Its vigorous built quality makes you more comfortable and safe during your ride. Toyota Etios Liva is also known to be the best taxi vehicle in the present services. It's spacious and contains all the features you need. All you need to do is hire Toyota Liva, sit back and enjoy your ride.
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Its D-4D oil burner is based on a SOHC valve configuration with 4-cylinders and 8-valves . This diesel engine is able to generate 67.06bhp and 170Nm of Torque output. On the other hand its petrol engine can generate 78.9bhp and 104Nm Torque output. The diesel engine is of 1364 cc and petrol is of 1197 cc.
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The interior of Etios Liva is amazing and different from all other cars. This car has odometer in center and other cars has odometer in the right side which is mainly visible to the driver only but in Etios Liva all the passengers can see speed of the car in the odometer and other features in it. There is lot of space in the car. It can easily hold 5 passengers in it including driver. The dashboard is of good quality. The HVAC of the car is very good as well as it has glove box cooling too in it. It has 12V power socket, adjustable headrests and height adjustable driver's seat. It has 251 litre boot compartment and 45 litre fuel storage capacity in it.


With our new Convention Centre and the advent of digital technology, everything is now possible for your events. At Atlantia, state-of-the-art technology works hand-in-hand with your events, and La Baule Events works with its innovative partners to offer you creative digital applications and connected services dedicated to tomorrow's professional meetings.
Traditional or electrically –powered projectors, personalised gobos, digital sound, multi-casting et simultaneous translation…our Son et Lumière Light show will bring all these possibilities to your event.
Video today is an integral tool in event management. At Atlantia, we offer you a new technological range of services: video projector with 20,000 lumens, info-décor, video mapping, streaming and the possibility of video-conferencing in workshops.
Atlantia Convention Centre owns the stands and wall posters allowing the diffusion of scientific knowledge of its exhibitions in an ecologically responsible manner.
Our teams take care of setting up your exhibitions and shows and also provide the entire service: set-up, furniture, decoration, reception and storage of parcels, storing exhibition documents and pieces, registration management ….
We will custom-make the room's decor, stage, staircase or screens…we will decorate whatever you like, according to your taste and imagination!
We offer dynamic display with our 9 screens 48''HD with complete device programming allowing users to be directed and informed accordingly. Our digital capacity is further supported with totem A3 colour printouts for signage. For the façade of the Convention Centre, we can offer you our 150 sqm video wall which we can personalise for you using the modular window panels.
With the possibility of up to 1,000 simultaneous broadband WIFI connections, stay connected all through your event and offer your participants a unique experience! Allow your guests to mix before, during and after the event. Make your room interactive, stage your Plenary session and captivate your public (we can provide tablets). Encourage networking and measure your event's success! You can also opt for individual WIFI connection through our own 4G hotspot.
To give your participants a lasting souvenir and make your event the talk-of-the-town, install a Selfie photo or video booth or get them to try a brand-new virtual reality by teleporting your guests to infinity!


In addition to tempo travellers, Gozo provides various services in Fatehabad including one way taxi drops or short roundtrips to nearby cities. You can also book cabs for your vacation or business trips for single or multiple days within or around Fatehabad or to nearby cities and towns.
With Gozo you can book a one way cab from Fatehabad to Bhuna, Daroli, Sardulgarh, Sirsa, Boha . Gozo provides well maintained air-conditioned (AC) cars, courteous drivers and our services are available 24x7x365. Gozo uses local taxi operators in Fatehabad who maintain highest level of service quality and have a very good knowledge of the local roads and highway driving. We also provide local sightseeing trips and tours in or around the city. Our cars and drivers serve tourists, large event groups and business people for outstation trips and also for local taxi requirements in Fatehabad.